Herbrand disjunction
E238239
Herbrand disjunction is a logical formula formed as a finite disjunction of ground instances of a first-order formula, central to Herbrand’s theorem in proof theory and automated reasoning.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Herbrand disjunction canonical | 4 |
How this entity was disambiguated
This entity first appeared as the object of triple T2139597 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Herbrand disjunction Context triple: [Jacques Herbrand, developedConcept, Herbrand disjunction]
-
A.
Entscheidungsproblem
The Entscheidungsproblem is a foundational decision problem in mathematical logic that asks whether there exists a general algorithm to determine the truth or falsity of any given first-order logical statement.
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B.
Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
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C.
Knuth–Bendix completion algorithm
The Knuth–Bendix completion algorithm is a procedure in term rewriting and automated theorem proving that transforms a set of equations into a confluent rewriting system, enabling decision of word problems in algebraic structures.
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D.
Church–Rosser property
The Church–Rosser property is a confluence property of rewriting systems stating that if an expression can be reduced in different ways, all reduction paths can be further reduced to a common equivalent form.
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E.
Jacques Herbrand
Jacques Herbrand was a French mathematician and logician known for his foundational contributions to proof theory and mathematical logic, particularly Herbrand's theorem.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Herbrand disjunction Target entity description: Herbrand disjunction is a logical formula formed as a finite disjunction of ground instances of a first-order formula, central to Herbrand’s theorem in proof theory and automated reasoning.
-
A.
Entscheidungsproblem
The Entscheidungsproblem is a foundational decision problem in mathematical logic that asks whether there exists a general algorithm to determine the truth or falsity of any given first-order logical statement.
-
B.
Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
-
C.
Knuth–Bendix completion algorithm
The Knuth–Bendix completion algorithm is a procedure in term rewriting and automated theorem proving that transforms a set of equations into a confluent rewriting system, enabling decision of word problems in algebraic structures.
-
D.
Church–Rosser property
The Church–Rosser property is a confluence property of rewriting systems stating that if an expression can be reduced in different ways, all reduction paths can be further reduced to a common equivalent form.
-
E.
Jacques Herbrand
Jacques Herbrand was a French mathematician and logician known for his foundational contributions to proof theory and mathematical logic, particularly Herbrand's theorem.
- F. None of above. chosen
Statements (30)
| Predicate | Object |
|---|---|
| instanceOf |
concept in automated reasoning
ⓘ
concept in proof theory ⓘ logical formula ⓘ |
| appearsIn | proofs of first-order validity via Herbrand's theorem ⓘ |
| constructedFrom |
instances of the matrix of a Skolemized formula
ⓘ
substitutions of ground terms for variables ⓘ |
| formalizationLanguage | classical first-order logic ⓘ |
| hasComponent |
finite disjunction
ⓘ
ground instances of a first-order formula ⓘ |
| hasDomain |
Herbrand universe of the underlying language
ⓘ
linked to:
Herbrand universe
|
| hasOppositeConcept | Herbrand conjunction (for universal formulas) ⓘ |
| hasProperty |
contains no variables (is ground)
ⓘ
is a finite disjunction ⓘ is built from instances of a given first-order formula ⓘ |
| isCentralTo | Herbrand's theorem ⓘ |
| isDefinedInContextOf | first-order logic ⓘ |
| namedAfter | Jacques Herbrand ⓘ |
| occursIn |
Herbrand-style proof calculi
ⓘ
linked to:
Herbrand's theorem
cut-elimination style arguments involving Herbrand's theorem ⓘ |
| relatedTo |
Herbrand base
ⓘ
Herbrand expansion ⓘ Herbrand universe ⓘ |
| roleInHerbrandTheorem |
serves as a propositional counterpart of a first-order formula
ⓘ
witnesses the unsatisfiability of a first-order formula ⓘ |
| usedFor |
reducing first-order validity to propositional validity
ⓘ
search procedures in automated theorem proving ⓘ semantic characterization of first-order consequence ⓘ |
| usedIn |
automated reasoning
ⓘ
automated theorem proving ⓘ proof theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Herbrand disjunction Description of subject: Herbrand disjunction is a logical formula formed as a finite disjunction of ground instances of a first-order formula, central to Herbrand’s theorem in proof theory and automated reasoning.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.